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Full text of "Treatise On Analysis Vol-Ii"

272        XIV    INTEGRATION IN LOCALLY COMPACT GROUPS

is integrable with respect to the product measure fa  u2  *  *  \in on Gn. It
follows immediately from this definition and from (13.21.17) that the sequence
G")i^i^ is convolvable if and only if the sequence (|^il)i^^w of absolute
values of these measures is convolvable. Moreover, it is clear that

J-J

H\ (xn)

is then a positive linear form on JTR(G) and therefore a positive measure on
G (13.3.1) ; and for all/e JfH(G) we have

(14.5.1 )

I f    f

by virtue of (13.16.5) and (13.21.17). It follows directly (13.1.1) that
/H> f    \ f(x1x2 ' ' ' xn) d^i(#i)   * d^nCO is also a (complex) measure on G.
This measure is denoted by ut * ^2 *  - - * un and is called the convolution pro-
duct or convolution of the sequence (u^ ^t^n- The formula (1 4.5.1 ) also shows
that

(14.5.2)               I/*! * /*2 * -   * 0J ^ K| * |/i2| *    * |/ij.

For each function/e 5fc(G), the function (xl9 . . . , ^n)i-^/(jc1A:2    xn) is
continuous and therefore measurable with respect to every measure on Gn;
hence, by virtue of (13.21 .10), the sequence (jii9 . . . , nn) is convolvable if and
only if

(14.5.3)

f *d |^(1)| (x,(1))   \

4- oo

for some permutation a of {1, 2, ...,}. An equivalent condition is that, for
each compact subset K of G, the (closed) set A c G" of points (xl9 x2,..., xn)
such that xtx2   * JCB e K is (^  ju2      ^-integrable.

If a sequence (/z, v) of two measures is convolvable, we say that ^ and v
(in this order) are convolvable, or that \i is convolvable on the left with v, or
that v is convolvable on the right with u. If ^ and v are convolvable and /z', v'
are two measures such that |ju'| ^ |ju| and |v'| ^ |v|, then it is clear from
(14.5.3) that n' and v' are convolvable. measures